Problem 255 - Intermediate Algebra

Question

In the following exercises, add or subtract. $$ 2.51-7.4 $$

Step-by-Step Solution

Answer

-4.89

Step 1: Align the Numbers

Align the numbers vertically by their decimal points: \[ \begin{array}{r} 2.51 \ -7.40 \end{array} \]

Step 2: Prepare for Subtraction

Notice that subtracting a larger number from a smaller number yields a negative result.

Step 3: Subtract the Decimal Portion

Perform the subtraction on the decimal portion first. Subtract 0.40 from 0.51, borrowing if necessary: \[ 0.51 - 0.40 = 0.11 \]

Step 4: Subtract the Whole Number Portion

Subtract the whole number portion. Since you are subtracting a larger whole number (7) from a smaller one (2), you expect a negative result: \[ 2 - 7 = -5 \]

Step 5: Combine the Results

Combine the results of the whole number and decimal parts: \[ 2.51 - 7.40 = -4.89 \]

Key Concepts

Aligning Decimal Points

When performing decimal subtraction, it's essential to align the numbers by their decimal points. This means each digit should be in the correct column, with the decimal points directly above one another. For example, if you are subtracting 7.4 from 2.51, you would write it as:
2.51
- 7.40
This alignment helps in arranging the numbers properly and makes the subtraction process straightforward.

• Make sure all the decimals are in line before performing any subtraction.
• If necessary, add trailing zeros to make the numbers easier to handle.

Borrowing in Subtraction

Borrowing is an essential concept in decimal subtraction. It occurs when the digit being subtracted is larger than the digit you are subtracting from. Let's look at the decimal portion of our example:
2.51 - 7.40.

Subtracting the Decimal Portion:

Begin with the decimal part: 0.51 - 0.4. Since 0.51 has a higher decimal value, this part is straightforward: 0.51 - 0.40 = 0.11.
However, if borrowing were necessary (as in 2.5 - 7.4), you would reduce one whole unit from the integer part and add 10 to the corresponding decimal figure.

• Reduce the next higher digit by one and add ten to your current digit.
• Subtract as usual now that you have borrowed the necessary amount.

Negative Results in Subtraction

Negative results occur when a smaller number is subtracted from a larger number. Notice in our problem, 7.4 is larger than 2.51. Subtracting 7.4 from 2.51 will naturally result in a negative number. You can think of this in terms of owing more than you have.
In the example:

Subtracting the Whole Number Portion:

Perform the operation on the whole numbers: 2 - 7 = -5. Since 2 is less than 7, the result is negative.

• Remember, negative results show that the number being subtracted was larger.
• Use parentheses to denote negative values for clarity.

Combine this with the decimal portion results: 0.11. Thus, the final result is: 2.51-7.40 = -4.89